Involution Products in Coxeter Groups

Abstract

For $W$ a Coxeter group, let $\mathcal{W} = \{ w \in W \;| \; w = xy \; \mbox{where} \; x, y \in W \; \mbox{and} \; x^2 = 1 = y^2 \}$. If $W$ is finite, then it is well known that $W=\mathcal{W}$. Suppose that $w\in \mathcal{W}$. Then the minimum value of $\ell (x)+\ell (y)-\ell (w)$, where $x,y\in W$ with $w=xy$ and $x^2=1=y^2$, is called the \textit{excess} of $w$ ($\ell$ is the length function of $W$). The main result established here is that $w$ is always $W$-conjugate to an element with excess equal to zero.

This is the preprint version. We also include, on the final page, a short Corrigendum correcting an error in Theorem 1.1. We are grateful to the referee of a later paper for pointing this out. The Corrigendum appeared as "Corrigendum to Involution products in Coxeter groups [J. Group Theory 14 (2011), no. 2, 251--259]" in J. Group Theory 17 (2014), no. 2, 379--380